Real Analysis 47 | Proof of Taylor's Theorem
Вставка
- Опубліковано 12 лис 2024
- 📝 Find more here: tbsom.de/s/ra
👍 Support the channel on Steady: steadyhq.com/e...
Other possibilities here: tbsom.de/sp
You can also support me via PayPal: paypal.me/brig...
Or via Ko-fi: ko-fi.com/theb...
Or via Patreon: / bsom
Or via other methods: thebrightsideo...
Join this channel on UA-cam: / @brightsideofmaths
💬 Access to the community forum: thebrightsideo...
🕚 Early access for videos: thebrightsideo...
❓ FAQ: thebrightsideo...
🛠️ What tools do you use: thebrightsideo...
Please consider to support me if this video was helpful such that I can continue to produce them :)
Each supporter gets access to the additional material. If you need more information, just send me an email: tbsom.de/s/mail
Watch the whole video series about Real Analysis and download PDF versions, quizzes and exercises: tbsom.de/s/ra
Supporting me via Steady is the best option for me and you. Please consider choosing a supporter package here: tbsom.de/s/sub...
🌙 There is also a dark mode version of this video: • Real Analysis 47 | Pro...
🔆 There is also a bright mode version of this video: • Real Analysis 47 | Pro...
🔆 To find the UA-cam-Playlist, click here for the bright version: • Real Analysis
🌙 And click here for the dark version of the playlist: • Real Analysis [dark ve...
🙏 Thanks to all supporters! They are mentioned in the credits of the video :)
This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.
x
#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives
I hope that this helps students, pupils and others. Have fun!
(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
For questions, you can contact me: steadyhq.com/e...
I found your channel by chance, and I was simply amazed by the quality and the consistency. Keep up the good work!
Thanks! Welcome aboard!
Thank you so much for this video!
I couldn't get where the derivitive of the auxiluary function comes from, but you've shown it so plainly and explained it so easily!
Thanks a lot :)
Countless Physics and Engineering Majors saved from intractable problems!
I'm happy to finally be able to understand the proof for Taylor :)
Thank you very much for this❤️
The way you prove these theorems is similar to creating an art work. I do not have any clue how you come up with these neat proofs! I hope at some point I can also be able to come up with proofs by myself. At this point I only follow your beautifully explained ones.
lol he didnt come up with all the proofs in analysis
Always such a joy to see a new real analysis video! Thank you for the lessons!
Excellent ,
very nice, short, and instructive proof
Many thanks!
I passed analysis 1 year ago but I Still love ur videos
Nicely done
Congratulations you are a real very good teacher we are looking forward expanding your video playlists thanks alot
Thank you! 😃
Really nice proof!
At 5:08, the t and x0 switched places in the definition of capital F subscript n, h
Yes, that is a mistake there. Sorry.
Thanks!
Great video as usual!
Quick question: is it sufficient that the remainder of a Taylor polynomial up to 2nd order includes h^3 to prove that it is the best quadratic approximation?
If not, how do I prove that it is the best quadratic approximation to a function?
The 2nd order polynomial is the best quadratic approximation in the sense described in the video. So it's already proven. What exactly do you want to show now?
@@brightsideofmaths is that 2nd order polynomial the best because the remainder is of a higher order?
Yes, the remainder goes faster to zero as stated when x approaches x_0. This is actually what we mean by best quadratic approximation in this context.
At 2:31, the professor said “when k is equal to zero, this factor here is defined to be 1.” Is the professor also saying that (h + x_0 - (x_0 + h))^0 is equal to 1 by definition? Do we define 0^0 to be 1 in this series? I thought 0^0 is usually not defined
We don't define 0^0 but the symbol x^0. In other words, we write x^0 but mean the constant function 1.
Thank you! So Taylor expansion does not guarantee that the approximate polynomial $T_n(h)$ is the best fit, i.e. some difference like $|f(x+h)-T_n(h)|^2$ is minimized! Then why is it the best fit?!
It's the best fit in the sense of the remainder term around the expansion point x_0.
Polynomials that minimize a certain norm are in general different from a Taylor expansion. So if you want a "best fit" polynomial over a finite interval, Taylor expansions are usually not a good bet.
But they are an invaluable tool for all kinds of convergence analyses or analyses around a small neighborhood of a point.
Tut mir leid, wenn das der falsche Platz für so eine Frage ist, aber du hast einen Kommentar geschrieben, dass du das Boox Note Air 2 für Mathe nutzt. Ich bin Mathestudent und spiele schon länger mit dem Gedanken mit ebenfalls ein E-Ink Tablet zu kaufen. Das Remarkable 2 gab es für 260 Euro und so musste ich zuschlagen. Jetzt meine eigentliche Frage, geht es dir generell um das Schreiben auf so einem E-Ink Tablet oder hat das Note Air 2 besondere nützliche Eigenschaften die es besonders gut für mathematisches macht, dass du so begeistert bist? Danke.
Danke für die Frage! Das Remarkable 2 ist mit Sicherheit genauso gut für deinen Anwendungsfall. Letztendlich hat mich das Note Air 2 überzeugt, da ich alle PDFs problemlos darstellen kann und direkt Kommentare dazuschreiben kann :)
@@brightsideofmaths Danke dir :)
First time I understood
Are you going to be covering integration proofs in the future? Very nice videos btw
The next videos will be about integration :)
@@brightsideofmaths brilliant stuff :)
Can you help me
The author must be a mathematician or a PhD in mathematics at least
I want proper statement of this proof
Proper in which way?
ngl that capital F looks alot like a T...
I see the problem but fortunately I chose another colour there :D